I believe to have understood that some physical experiments suggest finite values to divergent series (please correct me if I'm wrong, my understanding of these matters is limited).
I heard, for example, that the "equality" $$ \sum_{n=1}^{ \infty} n = - \frac{1}{12} $$ was suggested by some experiment conducted by physicists.
I was wondering if there are experiments in physics that seem to suggest two or more different values to the same divergent series. If not, why is this the case?